Fiber modes carrying orbital angular momentum (OAM) have recently received considerable attention in connection with increasing channel capacity in optical communications [1–3]. Several approaches to efficiently excite such modes have been reported [4–7] and circularly symmetric fiber designs have been used to maintain high fidelity OAM order during transmission.

In recent years there has been considerable interest in helicoidal fiber structures in connection with, for example, elimination of higher order modes in large-mode-area fibers [8], chiral fiber gratings [9], and spin–orbit coupling [10]. Helically twisted photonic crystal fiber (PCF) has been shown to support leaky OAM cladding modes that phase match to the core light at certain wavelengths, causing dips in the transmitted spectrum [11]. At wavelengths between these dips, the twisted PCF displays optical activity [12,13].

Here we report that a continuously twisted PCF with a novel three-bladed core lifts the degeneracy between left- and right-spinning modes of the same OAM order. It is, therefore, able to inhibit scattering between these modes—something that is very difficult if not impossible to achieve in circularly symmetric fibers or untwisted sixfold-symmetric hollow-core PCFs [14]. Although fibers capable of preserving linear polarization states have been available for several decades, and cylindrically birefringent PCFs have recently been demonstrated [15], this is the first report to our knowledge of a fiber that can robustly preserve the sign of the OAM.

How is it that a fiber with a threefold symmetric core can support OAM modes? We show, using a newly developed semi-analytical theory, that azimuthal Bloch waves are able to circulate unimpeded around the core, creating an OAM mode when a resonant condition is satisfied. Clockwise and anticlockwise versions of these modes become nondegenerate when the fiber is twisted, rendering them resistant to external perturbations. The analysis produces results that agree remarkably well with full-vectorial finite element modeling (FEM).

Figures 1(a) and 1(b) show a schematic of the core structure and a scanning electron micrograph of the silica–air PCF used in the experiments. It was fabricated using the stack-and-draw technique [16], the three-bladed Y-shaped core being constructed by replacing four capillaries with solid rods in the preform stack. In the drawn fiber, the hollow channels had a diameter of $\sim 1\mathrm{\mu m}$ and were spaced by $\sim 3\mathrm{\mu m}$; the outer diameter of the fiber was 116 μm. A continuous permanent twist (twist rate $\alpha =2\pi /L$, where $L$ is the helical period) was induced in the fiber by thermal postprocessing using a ${\mathrm{CO}}_2$ laser as heat source [11,17,18].

 

Fig. 1. (a) Schematic of the fiber core structure. (b) Scanning electron micrograph of the fiber used in the experiments. (c) Axial Poynting vector distribution of one of the ring modes at a wavelength of 800 nm and a twist rate of $1.26\mathrm{rad}/\mathrm{mm}$, calculated by finite element modeling in a helicoidal reference frame. (d) Axial Poynting vector distribution of the lowest order mode of the same twisted structure, which does not carry OAM.

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A scalar eigenmode analysis of the untwisted fiber reveals that there are always two degenerate eigenmodes involving pairs of satellite cores (see Supplement 1). These can be superimposed to produce degenerate ring modes of OAM order $\ell =\pm 1$, in which the phase advances by $\pm 2\pi /3$ from core to core.

Full vectorial FEM solutions of Maxwell’s equations in a helicoidal reference frame [13,19] show that four nondegenerate versions of these ring modes exist in the twisted fiber. All four have very similar Poynting vector distributions [Fig. 1(c) plots an individual profile for a twist rate of $1.26\mathrm{rad}/\mathrm{mm}$], and for twist rates larger than $0.1\mathrm{rad}/\mathrm{mm}$ they are almost perfectly (within 3%) left- (LC, spin $s=+1$) or right- (RC, spin $s=–1$) circularly polarized. Since the phase is undefined at the center, the intensity profiles exhibit a central null.

The simulations also reveal that the lowest order mode is strongly concentrated in the central core [Fig. 1(d)] and is optically active, supporting nondegenerate LC and RC polarized modes with circular birefringence $B\sim b\times \alpha$, where $\alpha$ is in rad/mm and $b=5\times {10}^{-7}\mathrm{mm}/\mathrm{rad}$.

Knowing that ring modes exist in the structure allows us to construct a simple but highly instructive approximate model for the twisted system. We begin by considering a generic helical and azimuthally periodic structure, described mathematically by its dielectric constant distribution:

A Bloch wave in general is written as the product of a perfectly periodic function and a phase progression at a rate given by the Bloch wavevector $k_B$ [20]. The fields that fit into the twisted structure can be written in a modified Bloch wave form, once it is recognized that the periodic function that makes up the Bloch wave must rotate at the same rate as the structure. Ignoring the radial dependence (i.e., treating the structure as a thin helical shell of radius $\rho$), a suitable Ansatz is, therefore,

Inserting Eqs. (1) and (2) into Maxwell’s equations in cylindrical coordinates with no radial term,

The OBMs are free to propagate (with a constant azimuthal group velocity component) around the circle in either direction, encountering effectively an infinite sequence of periods. The closed nature of the azimuthal path means, however, that only certain discrete values of $k_B$ are permitted, which leads directly to the resonance condition $k_B^{\ell }=\ell /\rho$, where $\ell$ is the OAM order. Once this condition is satisfied for the lowest order harmonic, it is automatically satisfied for all the others, with the difference that the OAM order of the $n$th harmonic is ${\ell }_n=\ell +3n$.

With this information, Eq. (4) can be rewritten as the equation set

Figure 2 shows the wavevector diagram for the experimental parameters (wavelength 800 nm, twist period 5 mm, radius $\rho =3\mathrm{\mu m}$) plotted over four Brillouin zones. OBMs form at $\ell =\pm 1$, as marked by the vertical dashed lines. The OBM group velocity (given by $\partial \omega /\partial \mathbf{k}$, where ω is the optical angular frequency and k the wavevector) points normal to the curves [20], so that the solutions for the $\ell =+1$ mode rotate toward the right, whereas those for the $\ell =-1$ mode rotate toward the left (note that the scales on the horizontal and vertical axes are not the same, so that the local gradient of the curve is exaggerated). Of crucial interest here, however, is that the propagation constants of the two modes are different, i.e., they are nondegenerate. The spectral content of each mode is also plotted in Fig. 2, showing the relative strengths of the four most significant OAM components.

 

Fig. 2. (a) Illustrative wavevector diagram for an OBM, plotted over four Brillouin zones. It was obtained by solving Eq. (4) with ${\beta }_0\rho =27.3$, $\alpha \rho =0.0038$, and $\kappa \rho =0.1$, including seven harmonics. Its shape is identical in each Brillouin zone (this ensures that every harmonic of the OBM shares the same group velocity—see text), and it is tilted at an average slope $\alpha \rho$ (this ensures that the field pattern rotates with the helical structure). (b) Normalized strengths of each harmonic. The numbers correspond to the harmonic orders $n$ of the backward-spinning (black) and forward-spinning (red) OBMs.

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For accurate comparison with experiment, and to assess the accuracy of the Bloch wave analysis, we employed a finite element model to calculate the values of propagation constant in the helicoidal reference frame. These must be transformed back into the laboratory frame using the relationship

The calculated effective refractive indices of the $n=0$ harmonic of these modes at 800 nm are plotted in Fig. 3(a) as a function of twist rate. The indices of the $\ell =+1$ OBMs increase with twist rate, while those of the $\ell =-1$ OBMs fall. The index splitting is proportional to the twist rate, as may also be seen in Fig. 3(a). The results of the scalar analysis agree very well with the full numerical simulations. To obtain these fits, the following parameter values were used: $\kappa \rho =0.078$, $\rho =3.08\mathrm{\mu m}$, and ${\beta }_0\rho =34.144$.

 

Fig. 3. (a) Comparison of FEM and analytical model. Left-hand axis: calculated refractive indices $n_{\mathrm{OBM}}$ at 800 nm of the $n=0$ harmonic of the $\ell =+1$ and $–1$ OBMs, plotted versus twist rate. The dots correspond to full vector finite element modeling and the solid lines are solutions of the analytical OBM model. The index of the $n$th Bloch harmonic is $n_{\mathrm{OBM}}-3n\alpha /k$. LC and RC polarized OBMs have refractive indices that differ by $\sim 3\times {10}^{–6}$ [see (b)]. Right-hand axis: index splitting between $\ell =+1$ and $–1$ OBMs, plotted versus twist rate. Within the narrow shaded region on the left, the $(s,\ell )=(-1,+1)$ and $(+1,-1)$ modes are less than 97% circularly polarized (within the accuracy of the FEM, the other two OBMs are at least 99.9% circularly polarized at zero twist rate). (b) FEM calculations of the optical activity of each OBM, plotted against twist rate (see text).

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In addition to the large OAM splitting, there is a smaller nondegeneracy between LC- and RC-polarized modes (all four OBMs are more than 97% circularly polarized for twist rates greater than $0.1\mathrm{rad}/\mathrm{mm}$). FEM calculations of optical activity $B=|n_{\mathrm{OBM}}^{\mathrm{LC}}-n_{\mathrm{OBM}}^{\mathrm{RC}}|$ for each OBM are plotted against twist rate in Fig. 3(b). The Fano-like feature at $\sim 4.6\mathrm{rad}/\mathrm{mm}$ is caused by an anti-crossing between the $n=+1$ harmonic of the $\ell =-1$ OBM (total $\mathrm{OAM}=0$) and the non-OAM lowest order mode.

A schematic of the experimental setup is shown in Fig. 4(a). A diode laser with wavelength $\sim 800\mathrm{nm}$ was used as a light source. A quarter-wave plate was used to select either LC- or RC-polarized light before launching into the fiber. The length of twisted PCF used in the experiment was $\sim 5\mathrm{cm}$ with a twist rate of $\sim 1.26\mathrm{rad}/\mathrm{mm}$. The first four higher order modes could be independently excited by adjusting the launching conditions. To verify that these modes do indeed carry OAM, an interferometric technique was employed, similar to that commonly used for observing the phase structure of optical vortices [21]. The output from the twisted PCF was superimposed on an expanded Gaussian reference beam and imaged onto a CCD camera. The path difference between the two arms was controlled using an adjustable delay line in the reference arm. A spiral-shaped interference pattern is expected if the light from the fiber carries OAM. This is indeed the case, single-spiral interference patterns being observed, as shown in Fig. 4(b), which confirms the principal OAM order of the modes. The handedness of the spiral indicates of the sign of $\ell$. The experimental results are in excellent agreement with numerical simulations. Note that the visibility of the spiral fringes is not expected to be 100%, owing to the presence of the other Bloch harmonics (OAM order $\ell +3n$). The visibility will be reduced further by unavoidable excitation of non-OAM fundamental modes, concentrated in the central core [Fig. 1(d)].

 

Fig. 4. (a) Schematic of the experimental setup for generating and characterizing OAM states in the twisted PCF. LD, laser diode; BS, beam splitter. (b) Experimentally recorded (top) and calculated (bottom) interference patterns created when ring modes with different combinations of spin and OAM order interfere with an expanded Gaussian reference beam.

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When the same experiment was carried out using an untwisted PCF, only concentric circles were seen in the interference pattern, no matter how the polarization and input conditions were adjusted. Although in principle it should be possible, by careful design of the OAM launch beam, to excite an isolated OBM rotating in one direction, it is likely that slight imperfections in the azimuthal periodicity will scatter light between the degenerate $\ell =\pm 1$ OBMs.

To confirm these results, $\sim 50\mathrm{m}$ of twisted fiber with hole diameter of $\sim 1.6\mathrm{\mu m}$, hole spacing of $\sim 5.2\mathrm{\mu m}$, and pitch of 5 mm was produced in a fiber drawing tower. The fiber was wound on a spool of diameter 16 cm. Using a 1550 nm laser and a spatial light modulator (SLM) setup to synthesize a three-lobed pattern carrying OAM [14], a clear single-spiral interference pattern was observed at the fiber output for excitation of the $\ell =\pm 1$ OBMs. Preservation of the $\ell =\pm 1$ states may be further enhanced by increasing the twist rate.

Twisted PCF with a “three-bladed” core supports a new kind of guided helical Bloch mode consisting of a superposition of Bloch harmonics with OAM orders $\ell \pm 3n$, coupled together by the twist. Each of these harmonics has a different propagation constant, given by ${\beta }_0+{\gamma }_{\ell }-3n\alpha$, so that the OBM has multiple values of refractive index, just as is the case for Bloch waves in regular periodic media [21]. Two nondegenerate OBMs exist in the twisted fiber, with opposite signs of OAM. The splitting in modal index is proportional to the twist rate, and both LC and RC versions of each mode exist, with slightly different propagation constants. Full details of the twisted Bloch wave model, the implications of which we believe are quite wide-ranging, will be reported elsewhere. We note, finally, that many more higher order OBMs will exist in PCFs with rings of cores placed farther away from the axis, although a full understanding of these modes will require further detailed analysis. Since long lengths of twisted PCF can readily be produced in a fiber drawing tower, OBMs may be useful in increasing the number of channels in optical communications.